Triangulating Cappell-Shaneson homotopy 4-spheres

© 2017 Ahmad IssaThe smooth 4-dimensional Poincare conjecture states that if a smooth 4-manifold is homeomorphic to the 4-sphere then it is diffeomorphic to the standard 4-sphere. Historically, one of the most promising families of potential counterexamples to this conjecture is the family of Cappell-Shaneson homotopy 4-spheres. Over time, with difficult Kirby calculus computations, an infinite subfamily of Cappell-Shaneson homotopy 4-spheres was shown to be standard, that is, diffeomorphic tothe 4-sphere and hence are not counterexamples. More recently, Gompf showed that a strictly larger subfamily is standard using the fishtail surgery trick. In another direction, Budney-Burton-Hillman discovered a fascinating ideal triangulation of a smooth 4-manifold which they show is homeomorphic to a certain Cappell-Shaneson 2-knot complement X^4. They pose as a problem to show that it is in fact diffeomorphic to X. We solve this problem by directly triangulating X and observing that the triangulation we obtain is combinatorially isomorphic to theirs. In fact, we show that this triangulation is a layered triangulation. We then generalise our construction to show how to construct layered triangulations of once-punctured 3-torus bundles, generalising the well-known Floyd-Hatcher triangulations of once-punctured torus bundles. Next, we describe how we can use these triangulations of once-punctured 3-torus bundles to construct triangulations of Cappell-Shaneson homotopy 4-spheres. This involves understanding the Gluck twisting operation via triangulations. For some particular examples, we simplify the triangulations using Pachner moves to a standard triangulation of the 4-sphere. For these examples, this provides a new computational proof that the corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the 4-sphere